Peg solitaire is a
game for one player. You begin with an arrangement
of
pegs - the diagram below shows the most common starting
arrangement.
* * *
* * *
* * * * * * *
* * * * * *
* * * * * * *
* * *
* * *
The aim is to leave just one peg in the centre, by repeatedly jumping
one peg over another in a horizontal or vertical line, and removing the peg which was jumped over.
Two peg solitaire positions are sometimes said to be equivalent if one can be converted into the other by a sequence of forwards or backwards moves. (A backwards move is jumping a peg over an empty hole and filling the hole.) It can be proved that two positions are equivalent if and only if one can be transformed into the other by repeatedly taking any three adjacent pegs/holes and inverting them - replacing all the holes by pegs and all the pegs by holes. Since forwards and backwards moves are special case of this inversion procedure, it's clear that if two positions cannot be transformed into each other this way, then they are not equivalent. The inverse of this statement is not so obvious, but has been proved mathematically.